PHYS 391 – Fourier Transform Primer
Eric Torrence with modifications by Dean Livelybrooks
1
Introduction
This document is intended to give you just enough information to follow the discussion of
the experimental aspects of the discrete Fourier transform and related operations like the
power spectrum. This will not be a mathematically rigorous treatment of the Fourier series
or integral Fourier transform. A better description can be found in any decent textbook on
mathematical methods for physics.
The key concepts for this class are the experimental aspects related to the Nyquist criterion (sampling rates, etc.), aliasing, the need for windowing your data, and the relationship
between the amplitude or power spectrum and the raw Fourier transform. Without some
mathematical foundation, however, these concepts will be meaningless.
Parts of this write-up were derived from “Mathematical Methods for Physicists” by Arfken, parts of the experimental description were motivated by a similar treatment in “Advanced LabVIEW Labs” by Essick, and parts came directly from Wikipedia.
2
Fourier Series
In 1807, Baron Jean Baptiste Joseph Fourier asserted that any arbitrary function defined
over an interval (−π, +π) could be represented by the series
∞
∞
X
a0 X
bn sin nx
an cos nx +
+
f (x) =
2
n=1
n=1
(1)
where an and bn are the so-called Fourier coefficients for the function f (x). This work came
out of studies of heat distributions between two plates, but as a mathematician, natural
philosopher, and all-around bright guy (he also is credited with discovering the greenhouse
effect in the atmosphere) he realized the broad applicability of this sort of series expansion.
Many of Fourier’s contemporaries, including people as notable as Lagrange himself,
thought Fourier was nuts, but with time and a few preconditions it has been shown that
Fourier was essentially correct. One of the more important (mild) conditions on f (x) shown
by Dirichlet is that the function must only have a finite number of discontinuities, as well
as a finite number of minima and maxima in the range (−π, +π). Note that in particular
there is no restriction that f (x) has to be continuous (making the Fourier series much more
useful than the Taylor series) nor does f (x) in any way need to be periodic.
2.1
Fourier coefficients
One obvious property of the Fourier series is a direct separability into even terms where
f (−x) = f (x) defined by the cosine terms with coefficients an and odd terms where f (−x) =
−f (x) defined by the sine terms with coefficients bn . It follows then that a purely even
function will end up being described by non-zero an coefficients, while a purely odd function
1
will end up being described by non-zero bn coefficients. The a0 /2 term, while even, is also
special as it represents a constant term giving the average value of f (x) across the interval.
For all other terms, the average value across the interval is zero by construction.
The Fourier coefficients an , bn can be determined by the Sturm-Liouville Theory to be
Z
1 +π
an =
f (x) cos nx dx
(2)
π −π
Z
1 +π
f (x) sin nx dx.
(3)
bn =
π −π
Note that by convention the coefficient b0 does not exist (or is always zero, depending on
your point of view).
2.2
Examples
A trivial example of the Fourier series is the function f (x) = A cos 2x. From inspection, we
can see that the coefficient a2 = A while all other coefficients are zero. We can also see this
by direct application of Equation 3 as
Z
1 +π
an =
A cos 2x cos nx dx.
π −π
The determination of coefficients an , bn exploits a property of sinusoidal functions with integer
frequencies called orthogonality. The integral above integrates to zero for all n 6= 2 and gives
a2 = A for n = 2. Consider the explicit case of n = 3. Figure 1 shows the function
f (x) = cos 2x cos 3x over the range of the integral, and indeed this does evaluate to zero.
Compare this to the case where n = 2 shown in Figure 2 and it is clear that the integral of
this function will not be zero.
A better example of the application of the Fourier series is the step function described
by
(
0, −π < x < 0,
f (x) =
h, 0 < x < π.
The constant term can be seen directly to be a0 = h since the average value of f (x) across
the complete interval is h/2. Meanwhile it can be shown that the other even terms are all
zero as
Z
1 +π
an =
h cos nx dx = 0.
π 0
Note the change in limits of integration since f (x) is only non-zero for values of 0 < x < π.
The odd terms can then be found by integration to be
Z
h
1 +π
bn =
h sin nx dx =
(1 − cos nπ),
π 0
nπ
which can be simplified to give the result
(
bn =
2h
,
nπ
0,
if n is odd,
if n is even.
2
(4)
1
0.8
0.6
0.4
f(x)
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
−4
−3
−2
−1
0
1
2
3
4
x
Figure 1: The function f (x) = cos 2x cos 3x. The integral of this function is zero over the
range −π → π.
1
0.8
0.6
0.4
f(x)
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
−4
−3
−2
−1
0
1
2
3
4
x
Figure 2: The function f (x) = cos2 2x. The integral of this function is π over the range
−π → π.
3
The complete Fourier series, then, for the step function is
h 2h sin x sin 3x sin 5x
f (x) = +
+
+
+ ... .
2
π
1
3
5
(5)
The first four terms of this Fourier series are shown in Figure 3.
1.2
1
0.8
0.6
f(x)
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−4
−3
−2
−1
0
1
2
3
4
x
Figure 3: First four terms of the Fourier series describing a step function with h = 1.
2.3
Discussion
There are a couple of immediate observations to be made here. A Fourier series is a summation of fixed-frequency sine or cosine functions with argument nx. As such, the coefficients
describe the amplitude of the component with natural frequency n. For the step function, the series converges very slowly, which implies that there are significant high-frequency
components necessary to accurately describe the function. This is generally true with any
discontinuous function, or other function with “sharp” features. A function which is very
rounded and sinusoidal converges quickly in the Fourier series, while something like the step
function or triangle wave needs many many high-frequency components.
2.4
Exponential Form
An alternative, more compact representation of the Fourier series can be made by using
Euler’s formula
einx = cos nx + i sin nx.
(6)
Using this we can rewrite Equation 1 as
f (x) =
∞
X
n=−∞
4
cn einx ,
(7)
where we now have complex coefficients cn with the index n having both positive and negative
values. The coefficients can be seen to follow
(
1
(an − ibn ), n > 0,
cn = 21
(8)
(a + ibn ), n < 0,
2 n
while c0 = 21 a0 . Clearly the positive and negative coefficients are not independent, but rather
are complex conjugates of each other as cn = c∗−n . Now the real part of cn is related to the
even part of the function f (x), while the imaginary part of cn is related to the odd part.
3
Extended Fourier Series
So far we have only discussed the application of Equation 1 to some arbitrary function on
the range (−π, +π). Now we want to look at extending this to other situations.
3.1
Periodic functions
Nothing in the definition of Equation 1 requires the function f (x) to be periodic. In fact,
in our solution for the step function it was never specified what happened to the function
beyond the range (−π, +π). It should be clear, however, that since Equation 1 is formed
from sine and cosine functions which are periodic with period 2π, the resulting Fourier series
found on the defined range is also periodic outside that range with period 2π. As such, to
represent a continuously periodic function, one simply needs to find the Fourier coefficients
for a single period, and the solution with automatically also represent the solution over the
entire range of x. This technique can clearly not be used to represent a function with a
period greater than 2π, however.
3.2
Change of Interval
To apply equation 1 to a periodic function does require the function f (x) to be periodic over
an interval with length 2π. Clearly a simple change of variables can be applied to create
,
instead a transform applicable over a range 2L. By replacing n with the scaled quantity nπ
L
any range can be accommodated. The Fourier series, then, is written as
∞
∞
a0 X
nπx X
nπx
f (x) =
+
an cos
+
bn sin
2
L
L
n=1
n=1
(9)
with coefficients determined by
Z
1 +L
nπx
an =
f (x) cos
dx
L −L
L
Z
1 +L
nπx
bn =
f (x) sin
dx.
L −L
L
(10)
(11)
In this formulation it is clear that the quantity nπ
is a measure of angular frequency, which
L
could be a spatial frequency when applied to some function of position f (x) or a temporal
frequency when applied to some function of time f (t).
5
3.3
Change of Basis
While we won’t get into this here, it is worth noting that the Fourier series defined with
sine and cosine functions is just one example of a Fourier-style expansion. Any complete
and orthogonal set of functions can be used to define a similar Fourier-style series. While
sines and cosines are convenient for describing plane-wave solutions and physical problems
like waves on strings, for many other geometries other functions, such as Bessel functions
or Legendre polynomials are more appropriate. There are equivalent Fourier-Bessel and
Fourier-Legendre series which are well known, and in general any function can be expanded
as a series of any other set of complete, orthogonal functions. This last point is in fact a key
tenant of quantum mechanics.
4
Fourier Integral
We have shown how the Fourier series can describe a function bounded on a range (−L, L),
or periodic with period 2L, but what about describing a continuous and non-periodic func2
tion such as our favorite Gaussian function f (x) = e−x ? One obvious approach is to use
). Identifying this as a frequency (spatial or temporal),
Equation 9 in the limit limL→∞ ( nπ
L
this limit is equivalent to making the frequency resolution of the Fourier series infinitesimal, or in other words turning the discrete frequency sum of Equation 9 into a continuous
frequency integral.
In this limit, the Fourier series becomes the Fourier integral written as
Z ∞
f (x) =
[a(ω) cos ωx + b(ω) sin ωx]dω,
(12)
0
where the Fourier coefficients a(ω) and b(ω) are now continuous real functions of the frequency ω. These coefficients are determined by
Z
1 +∞
f (x) cos ωx dx
(13)
a(ω) =
π −∞
Z
1 +∞
b(ω) =
f (x) sin ωx dx.
(14)
π −∞
As before, we could also write this in exponential form leading to a Fourier integral of form
Z +∞
f (x) =
c(ω)eiωx dω,
(15)
−∞
where again c(ω) is now a complex function of the frequency parameter ω, related to the
real functions a(ω) and b(ω) as before.
5
Fourier Transform
The Fourier transform is one of a class of integral transforms which converts a function
from one form to another. The Fourier transform is really just a restatement of the results
6
of the continuous Fourier integral described by Equation 15. More specifically, the Fourier
transform is the operation for finding the complex Fourier coefficients in the exponential
form. The Fourier transform of the function f (x) is then defined as
Z +∞
ˆ
f (x)e−iωx dx,
(16)
f (ω) ≡
−∞
such that the inverse Fourier transform, which is equivalent to the Fourier integral from last
section, is given by
Z +∞
1
f (x) =
fˆ(ω)eiωx dω.
(17)
2π −∞
1
here is a choice of convention, and some definitions place an equal factor of
The factor of 2π
√1 in front of both the Fourier and inverse Fourier transforms.
2π
Again the transformed function fˆ(ω) is functionally the same thing as the Fourier coefficients c(ω) from Equation 15. This function is complex, and again the real part of this
function represents the amplitude (as a function of frequency) of the even frequency components while the imaginary part represents the amplitude of the odd frequency components.
In physics, one often applies the Fourier transform to a function of time f (t) to produce
the transformed function fˆ(ω) as a function of natural frequency. As no information is gained
or lost in this transformation (as can be seen from the ability to recover the original function
through the inverse Fourier transform), the functions f (t) and fˆ(ω) are two equivalent ways
of looking at the same information. These are typically called the time domain and frequency
domain respectively.
5.1
Fourier properties
While we aren’t really going to use much of this, there are a few key mathematical properties
of the Fourier transform which are useful to understand.
5.1.1
Linearity
The Fourier transform is linear in the sense that for any complex numbers α and β, the
function
h(x) = αf (x) + βg(x),
leads to a Fourier transform given by
ĥ(ω) = αfˆ(ω) + βĝ(ω).
5.1.2
Translation
For any real number x0 and function h(x) = f (x − x0 ), the Fourier transform is given by
ĥ(ω) = e−iωx0 fˆ(ω),
where fˆ(ω) means the Fourier transform of the function f (x). This shows that the difference
between the Fourier transform of a sine or cosine function is simply a constant complex factor
(phase shift).
7
5.1.3
Modulation
For any real number ω0 and function h(x) = eiω0 f (x), the Fourier transform is given by
ĥ(ω) = fˆ(ω − ω0 ).
This is directly related to the process of amplitude modulation, where a signal function
f (x) is multiplied (modulated) by a fixed carrier frequency ω0 . The result in the frequency
spectrum is a variation in amplitude around the base carrier frequency. This is the dual
relation the the translation property shown above.
5.1.4
Scaling
For any real non-zero number a the Fourier transform of the function h(x) = f (ax) is given
by
1 ˆ
ĥ(ω) =
f (ω/a).
|a|
5.1.5
Differentiation
For the derivative of a function h(x) =
dn
f (x),
dxn
the Fourier transform is given by
ĥ(ω) = (iω)n fˆ(ω).
This result is one of the key useful properties of the Fourier transform, as it allows derivatives
in differential equations to be converted via the Fourier transform into a product of factors
iω and the Fourier transform of the original function.
5.1.6
Convolution
The convolution of two functions f (x) and g(x) is defined as
Z +∞
f (x − y)g(y)dy,
f (x) ⊗ g(x) ≡
−∞
which is often used to describe how detector resolution effects the observed (or recorded)
function compared to the true or input function which is typically what we really want to
measure.
The Fourier transform of a convoluted function h(x) = f (x) ⊗ g(x) is given by
ĥ(ω) = fˆ(ω)ĝ(ω),
or the simple product of the transformed functions.
Similarly, the dual relationship exists for the product of two functions h(x) = f (x)g(x)
such that
1 ˆ
ĥ(ω) =
f (ω) ⊗ ĝ(ω).
2π
8
f (x)
fˆ(ω)
eiω0 x
2πδ(ω − ω0 )
cos(ω0 x) π[δ(ω − ω0 ) + δ(ω + ω0 )]
sin(ω0 x) iπ[δ(ω + ω0 ) − δ(ω − ω0 )]
δ(x)
1
p
ω2
−αx2
e
π/α e− 4α
Table 1: Some common Fourier transforms
5.2
Specific Transforms
It is generally straightforward to evaluate Fourier transforms, but there are a few special
functions which are important enough to highlight here.
The first three entries in Table 1 are all variants of the usual interpretation of a Fourier
transform operation as picking out the frequency components of an input waveform. For a
sinusoidal function with frequency ω0 the transform fˆ(ω) is non-zero (Delta function) only
at the frequencies ω = ω0 .
The dual relation is the result that the transform of a Delta function (the most extreme
version of a “sharp feature”) is a constant, or in other words an infinite number of constant
terms across the entire frequency spectrum.
The last result shows another interesting feature of the Gaussian function. Under the
proper conditions (parameter α = 1/2), the Gaussian function is its own Fourier transform.
One final specific function needs some particular consideration, which is a rectangular
function
(
1
,
1, |x| < 2a
(18)
f (ax) =
1
0, |x| > 2a ,
which is a rectangular function centered at the origin with width 1/a. The Fourier transform
can be written as
Z +1/(2a)
fˆ(ω) =
e−iωx dx,
(19)
−1/(2a)
which with a little bit of manipulation leads to the result
1 sin u
fˆ(ω) =
,
|a| u
(20)
where u = ω/(2a). This is another example of a function with “sharp features” having a
large number of high frequency components in the Fourier transform.
6
Experimental Applications
A common use of the Fourier transform experimentally is to take some waveform, which
could be a audible sound signal or a voltage signal with respect to time V (t), and apply
the Fourier transform to the raw time-dependent function f (t) to extract information about
amplitude as a function of frequency fˆ(ω). For any problem which involves oscillations, this
9
is typically a much better way to visualize the data than to look at the time series directly.
Since we typically measure things as a function of time (although there is no reason we
couldn’t make equivalent measurements of something as a function of position) I am now
going to switch notation and start writing things naturally as a function of time.
Invariably, this experimental Fourier transform involves a first step of digitizing some
continuous analog waveform, also known as sampling. It doesn’t matter whether this is done
by a DVM, an ADC, an oscilloscope, or even a student with a stopwatch, the process of
digitizing a continuous function ends up approximating f (t) with the discrete time series
f (ti ) where discrete measurements are made at equally-spaced time intervals ti . The actual
value of f (t) also tends to be digitized into discrete values by the fixed-resolution nature of
an ADC, but that isn’t important for this discussion.
6.1
Nyquist Frequency
One of the first consequences of making a time series of discrete measurements f (ti ) comes
from the Nyquist-Shannon sampling theorem. For simplicity it is always assumed that the
sampling is done at a constant rate νs = 1/∆t known as the sampling frequency. For example,
if I take 1000 evenly-spaced samples over a 1 second period, I have a sampling frequency
νs = 1 kHz. Here I am reverting to the more experimentally-friendly unit of frequency
ν which is measured in cycles per second (or Hertz) rather than the theoretically-friendly
natural frequency ω = 2πν.
The sampling theorem says that for a given sampling frequency νs , the fastest real signal
you can theoretically hope to reconstruct accurately is given by the Nyquist (or critical)
frequency
νc = νs /2.
(21)
In other words, if you want to measure a signal with a frequency of 1 kHz, you will need to
sample this signal at a rate of at least 2 kHz. This is the reason why audio CDs are encoded
at a bit rate of 44 kHz. In order to accurately reproduce sounds through the range of human
hearing (up to 20 kHz) you need to have a Nyquist frequency which is slightly above 20 kHz,
which leads to a sampling frequency at twice the Nyquist or critical frequency.
6.2
Aliasing
An interesting (and potentially dangerous) experimental condition results from undersampling a signal, that is to say trying to measure a frequency ν which violates the Nyquist
condition such that ν > νs /2. The best example of this comes from considering a sampling
frequency which is just slightly below the actual signal frequency for a sinusoidal signal. If
the frequencies were exactly the same, the sampling frequency would always pick a point
of constant phase with respect to the signal, and the result would be a constant value every time. If the sampling frequency is slightly less than the signal by an amount ∆ν, the
sample will pick a point on the signal waveform which has been shifted by a small phase
∆φ = 2π∆ν∆t = 2π(νs − ν)/νs . The resulting sampled waveform, then, will be a slowly
varying function as each successive sample picks a point shifted by ∆φ which will have a
frequency not given by ν but rather will have an apparent frequency νs − ν. In the extreme
limit of ν = νs , the sampled signal will not appear to have any oscillation at all.
10
This effect is known as aliasing and with any discrete sample of data it is impossible to tell
the difference between an accurate measurement of a frequency and an aliased measurement
of a higher frequency back below the Nyquist frequency. As you will see in your lab, when
your signal is below the Nyquist frequency, you accurately measure that frequency. As the
signal exceeds the Nyquist frequency, the measured frequency seems to “turn around” and
you no longer measure ν but rather a frequency given by νc − ν which continues to decrease
as ν gets larger. As ν approaches νs , the apparent frequency goes to zero as described above,
then the process repeats itself.
For this reason, any careful experimental measurement of waveforms needs to make sure
high frequency components above the Nyquist frequency are filtered out (attenuated) such
that they don’t get aliased down below the Nyquist frequency and give erroneous measurements. This necessity for some “headroom” to do this low-pass filtering is exactly why CDs
are sampled at 44 kHz and not 40 kHz.
7
Discrete Fourier Transform
Since experimentally we have a discrete time series f (ti ) rather than a continuous function f (t), to perform a Fourier transform we need to replace the integral with a discrete
summation known as the Discrete Fourier Transform (DFT):
Xk =
N
−1
X
f (tj )e−i2πνk tj ∆t,
(22)
j=0
where now the integer number k identifies the complex discrete Fourier coefficient Xk , while
the index j is the summation index over the waveform, each spaced a distance ∆t apart.
7.1
Discrete Frequency Index
It is important to understand the similarities and differences between the discrete coefficients
labeled by k in Equation 22 and the discrete coefficients labeled by n in Equation 3. Here,
we have a discrete time series which, due to the nature of the Nyquist criterion, can only
give information on frequencies up a limit of to νs /2. In addition, and similar to Equation 3,
the finite nature of the interval gives rise to the result that the longest period the Fourier
expansion can reproduce is given by the length of the interval. Here, our interval is given by
N ∆t = N/νs . Another way of expressing this is that the smallest frequency we can resolve
is
∆ν = νs /N.
(23)
Clearly we can observe a component with a frequency of νk=0 = 0, but the next frequency
component then has a value given by ν1 = ∆νk and in general the frequency corresponding
to Xk is given by
νk = νs k/N.
(24)
Remembering that in the exponential form we need both positive and negative indexes k,
this naturally leads to a range of k running from −N/2, ..., −1, 0, +1, ..., N/2 and the largest
possible frequency being νs /2 just as the Nyquist criterion would tell us.
11
7.2
Frequency Resolution
The Discrete Fourier transform has an important but non-intuitive result buried in the
discussion of the frequency index. Because we have sampled a discrete number of points
spaced a distance ∆t = 1/νs apart, there is a minimum frequency difference that we can
resolve given by
∆ν = νs /N.
This can also be thought of as the frequency resolution as it is explicitly the smallest change
in frequency that we can resolve in our sampled waveform. Rather counter-intuitively, this
frequency resolution does not necessarily increase with a higher sampling frequency (remember, increasing frequency resolution corresponds with a decrease in ∆ν), in fact it gets worse,
as νs goes up if N remains fixed. To maintain a fixed interval of time (T = N ∆t = N/νs ), increasing the sampling frequency requires, also, linearly increasing the total number of points
(N ), and the frequency resolution ∆ν will remain unchanged. To increase the frequency resolution (reducing ∆ν) requires increasing N (total sample time) while keeping νs constant,
decreasing the sampling frequency νs while keeping N constant (which also increases total
sample time), or both.
So in other words, as long as your sampling frequency does not violate the Nyquist
criterion for the signal frequency you want to measure, the only way to improve the frequency
resolution is to sample for a longer time. Increasing the sampling frequency at best does
nothing, and if the total number of points N you can store is limited (as is often the case in a
buffered ADC) it will actually make the frequency resolution worse. Increasing the sampling
frequency is only necessary to make higher frequencies accessible, it does not in any way
improve the frequency resolution.
Clearly if you are trying to measure a transient signal (which implies lots of high frequency
components) you need a fast sampling frequency. The frequency resolution you can obtain,
however, will still be limited by the length of the pulse itself, regardless of your sampling
frequency.
7.3
Inverse Discrete Fourier Transform
While not so commonly used, to complete the story the inverse of the DFT operation, which
is the equivalent to the original Fourier series, is given by
xj =
N
−1
X
k=0
Xk i2πνk tj
e
,
N
(25)
which we can re-write in the more convenient form
xj =
N
−1
X
Ak ei2πkj/N ,
(26)
k=0
where we have explicitly expressed the complex amplitude component of frequency νk as
Ak = Xk /N.
12
(27)
It is important to note that in this convention, the coefficients returned by the DFT must
explicitly be divided by the total number of points N to obtain a properly normalized
amplitude.
7.4
Fast Fourier Transform
The most common term heard in regards to experimental uses of the Fourier transform is the
“Fast” Fourier Transform or FFT. The Discrete Fourier transform as written in Equation 22
is very slow to compute numerically, scaling as N 2 . The FFT (of which there are several
variants) is a specific algorithm for computing a DFT which uses certain symmetries in the
∗
problem (like the fact that Xk = X−k
) to speed up the execution time to at most N ln N .
This can be very significant for large data sets. One common feature enforced by most FFT
algorithms is that they assume the data sample size is some power of 2 such that N = 2n . If
a data sample is specified with N less than an even power of two, the series will be padded
with zeroes. This isn’t a big deal, but for maximal computational efficiency it is always good
to specify N as some binary power of two.
Some implementations of the FFT also assume that the input data series f (ti ) is realvalued. While there is no requirement for this to be true mathematically, since the FFT is
usually used on real experimental data, throwing out the imaginary part also reduces the
total amount of data to process by a factor of 2.
8
Using the FFT
Not surprisingly there is an fft function built into MATLAB. The input and output of
this function are both arrays of numbers which correspond to the operation described in
Equation 22. For most people, this is annoyingly far from the desired output of a histogram
of amplitude vs. frequency. Here we want to make the connection between the mathematical
object implemented in fft and the qualitative understanding of an FFT as a tool for finding
frequency components.
First, Equation 22 returns complex coefficients. Sometimes this is useful, but often we
are just interested in the total amplitude. One could either separate these values into their
real and imaginary components (which would show the even and odd functional components
separately)
p or probably more useful is to simply find the magnitude of each component
|Ak | = Ak A∗k . The absolute value function works as well. Alternately if you were really
interested in the relative phase of each frequency, you could find the phase of each component
Ak instead. In converting to amplitude, one must remember to include the factor of N as
shown in Equation 27, since the MATLAB function returns Xk directly.
Second, depending on the program, the array returned may correspond to the coefficients
Xk which run from k = −N/2, ..., −1, 0, 1, ...N/2 − 1 (total of N output coefficients). Computers don’t tend to like negative indexes, so often these are shifted so that the first entry
in the returned vector actually corresponds to k = −N/2. The constant term k = 0 comes
back in entry N/2 + 1, while the k value corresponding to the Nyquist frequency is only
represented once. Note that MATLAB does not do this, as you can easily see by taking the
fft of a series of constant values in MATLAB. MATLAB returns N coefficients where the
13
first entry is X0 with the remainder representing the values of Xk for positive k only. What
MATLAB does do (which is arguably more annoying) is returns values of k running from
0, 1, ..., N − 2, N − 1. The k = 0 term comes out first, but rather than just going to k = N/2
which is the Nyquist limit, it continues all the way to k = N − 1. If you plot this data you
∗
will see the entire spectrum reflected as you go past the Nyquist limit such that X1 = Xk−1
.
Since the first example is much more traditional in the output of numerical FFT algorithms,
MATLAB provides a function fftshift which takes the fft output and changes it into the
other form!
One has to be very careful to get the correct amplitudes from the repeated coefficients
(which are complex conjugates of each other). The amplitude of a cosine term, for example,
will be split evenly between the k > 0 and k < 0 coefficient. You need to count both
of these (or alternately multiply the magnitude of one half of the coefficients by 2) to get
the amplitude to properly work out. Annoyingly, this does not need to be done for the
k = 0 term. You have to be very careful with your vector manipulation if you want a
proper amplitude vs. frequency plot ranging from 0 < ν < νc since the constant term only
comes out once, as does the coefficient exactly at the Nyquist frequency. Multiplying the
two (complex conjugate) halves of the values over the range k = 1, N/2 − 1 will give you
directly the magnitude squared, which is directly equal to the Power spectrum, or take the
square root to get the amplitude spectrum. A less error-prone method is to simply truncate
the outputs before the Nyquist frequency and take the magnitude of the complex numbers
directly, doubling all but the zero-term.
Finally, to make a plot of amplitude vs. frequency, one also needs to know what frequencies each amplitude component corresponds to. This is not in the output of the fft
algorithm, in fact there is nothing in the arguments to the fft algorithm which says anything
about the ∆t or sampling frequency νs of the data. One must use Equation 24 to create a
vector of frequencies which can then be used as the horizontal axis for a vector of Fourier
amplitudes constructed as described above.
9
Windowing and Leakage
In addition to the problem of aliasing, there is a second experimental aspect to using the
FFT which we will explore in our lab which needs to be understood a bit theoretically. This
is the concept of Leakage and the necessity for Windowing your data.
9.1
Leakage
Consider what a FFT actually sees. For a discrete set of time (divided up into N specific
points) the function f (ti ) is defined, but outside that range absolutely nothing is known
about the function. A finite sampling of a continuous function, then, looks to the FFT
operation like a continuous function multiplied by a square rectangle such that outside the
sampling window the function is zero.
If we take the FFT of a pure cosine, then, which we would expect to give us a nice sharp
peak at the specific frequency of the sine wave, what we actually end up with is the FFT of
the function h(t) = f (t)g(t) where g(t) = cos 2πν0 t for example, and f (t) is the rectangular
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function we saw above
(
1, |x| <
f (ax) =
0, |x| >
1
,
2a
1
,
2a
(28)
We can use one of the Fourier transform rules to simplify ĥ(ν) to the convolution of
ˆ
f (ν) ⊗ ĝ(ν). For our pure cosine, we have the result that ĝ(ν) = δ(ν − ν0 ) + δ(ν + ν0 ) where
I have dropped the normalization constant for simplicity, and we must remember that ν can
be generally positive or negative. If we only care about the real magnitude, we can restrict
ourselves to positive frequencies, and we find just |ĝ(ν)| = δ(ν − ν0 ).
The Fourier transform of g(t) meanwhile, is given by the formula found in Equation 20
1 sin u
,
fˆ(at) =
|a| u
with the identification of 1/a = N/νs as the total sample length in time, and u = 2πν/(2a) =
πνN/νs = πν/∆ν. The convolution of a delta function ĝ(ν) = δ(ν − ν0 ) with the function
fˆ(ν) just results in shifting the original function by a distance given by ν0 so that ĥ(ν) =
fˆ(ν − ν0 ).
Lets think about this a moment. The sinu u function will naturally peak at a frequency
ν0 , and it will have zeroes spaced evenly away from this peak every time u changes by one
unit of π, or alternately every time ν changes by ∆ν. If we had infinite resolution on ν, we
would indeed see this long series of frequencies represented by the sinu u function (or at least
the magnitude thereof). Since we really only sample the amplitude as specific frequencies
given by νk , if we are lucky and ν0 happens to line up with one of the sampled frequencies νk
we will get one prominent spike, and all of the other frequencies will line up perfectly with
the zeroes of the sinu u function (since they are all exactly ∆ν away. If we are unlucky (more
likely) and ν0 does not line up with one of the sampled frequencies νk , what we see is going
to look a lot more like the sinu u function.
Another way of thinking about this is that the fact that we always sample a finite window
introduces higher order harmonics into our sample whether we like it or not. Our sample
always has some finite window applied to it, and if we do nothing, the edges of our window
are very sharp and lead to lots of higher-order frequency terms. This general feature is called
frequency leakage.
9.2
Windowing
As you will see in the lab, left uncontrolled, frequency leakage leads to some very nasty
effects. In particular, the total amplitude of a signal is going to remain constant, but if there
is frequency leakage, this amplitude will be split amongst all of the frequency components
and the apparent height of the central “spike” will vary up and down as the signal frequency
moves around. The width of the spike also varies similarly, making it hard to use the FFT
to measure either amplitude or width of a signal.
The solution is to apply a different window “by hand” which is much more smooth and
doesn’t contain the sharp edges which give rise to the high frequency components in the first
place. In practice you are trading off uncontrolled leakage with leakage with well defined and
stable characteristics. There are several common windows used, with names like Hamming,
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Hanning, Blackman, and Gaussian, with each defining a different mathematical function.
Without going into a lot of detail here, each has characteristics which makes it better for
measuring amplitudes, measuring frequencies, separating frequencies, or measuring spectral
widths. The Hamming window is probably the best general purpose window for finding
specific frequencies, as it gives a nice sharp spike without too much leakage. The Hanning
window (actually named for an Austrian named Hann, but colloquially called Hanning just
to confuse graduate students) tends to have less leakage, but in exchange for a broader
central spike. For a low amplitude frequency close to a high amplitude frequency, this can
be advantageous for resolving the smaller one, which otherwise might get lost in the tail
of a Hanning window. The Blackman window takes this to the extreme, having a minimal
amount of leakage, but in exchange for a very broad central spike. If there isn’t a necessity
of resolving two frequencies close together, however, this can be very useful.
Windowing is important, as you are windowing your data no matter what you do (with
a rectangular window, which is probably the worst choice of all)
10
Final Word
Much of the theory behind Fourier transforms and FFTs is (in my humble opinion) straightforward but not at all intuitive. The actual process of dividing a time series waveform into
frequency components is, however, one of the most powerful tools a physicist has at their
disposal when dealing with periodic (or even quasi-periodic) data. Obtaining a better intuitive understanding of the concepts described in this handout is the last goal of this course,
and the specific goal of your final lab assignment. The best use of your lab time is to just
play with the FFT application and try to make a connection between the theory presented
here and the way it actually behaves when looking at a real signal.
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